We know that countable Abelian groups are classified up to isomorphism by their Ulm-invariants.
In the course of research, the following questions have been raised for me. If anyone has an answer in this regard, I would be grateful if they could provide it to me.
Is there a countable reduced $p$-group such that the corresponding Ulm-Kaplansky invariants are infinite or zero?
In general, what sequences of ordinals can be Ulm-Kaplansky invariants of a countable reduced $p$-group?
These invariants are additive with respect to arbitrary direct sums. Hence, for any countable reduced abelian $p$-group $A$, the direct sum $A^{\oplus \mathbb{N}}$ is still countable and reduced with invariants all either $0$ or $\aleph_0$.
The more general question is the content of exercise 42 in Kaplansky's Infinite Abelian Groups (as well as several exercises building up to it). Let $\lambda$ be any countable ordinal and $f$ be any function from $\lambda$ to the cardinals up to $\aleph_0$. Then $f$ is the sequence of invariants for such a group of length $\lambda$ if and only if $f$ takes on infinitely many nonzero values between any two limit ordinals $\alpha<\beta\leq \lambda$.